This paper illustrates the results obtained from two-dimensional numerical simulations of multiple gas bubbles growing under buoyancy and electromagnetic forces in a quiescent incompressible fluid. A lattice Boltzmann method for two-phase immiscible fluids with large density difference is proposed. The difficulty in the treatment of large density difference is resolved by using nine-velocity particles. The method can be applied to simulate fluid with the density ratio up to 1000. To show the efficiency of the method, we apply the method to the simulation of bubbles formation, growth, coalescence, and flows. The effects of the density ratio and the initial bubbles configuration on the flow field induced by growing bubbles and on the evolution of bubbles shape during their coalescence are investigated. The interdependencies between gas bubbles and gas rate dissolved in fluid are also simulated. 1. Introduction A steadily increasing computational power, new developments, and improvement of numerical methods allow numerical simulation to model more and more physical phenomena. In this sense, the in situ gas generated by alumina particles and carbon engenders the formation of gas bubbles in aluminum cells. These gas bubbles are produced by chemical reaction of alumina with the carbon, which consists of complex process of electrochemical reduction of alumina in the cells. The continuously released gas of the alumina reaction generates nuclei underneath the surface of carbon anode and detaches to dissolve in the cryolite which subsequently grows to bubbles. The bubbles arrange into a cellular structure in the cryolite, which is preserved by a rapid cyclic flow of gas. A major aim of the simulations presented here is to gain a better fundamental physical understanding of gas bubbles because of the complexity of gas bubbles- liquid flow in chemical system [1, 2], since it determines largely the dynamic of the cryolite. The detailed understanding may help to reduce the energy consumption and to improve the present energy efficiency by controlling the dynamics of bubbles in cells. In contrast with actual aluminum electrolysis cell models, which focus rather on specific aspects and for the most part on gas bubbles system, a comprehensive and general gas bubbles model is needed to be developed here which includes the wealth of phenomena occurring in aluminum electrolysis cell process. The model will elucidate the influence of gas bubbles as described in [3, 4] and processing parameters such as the cryolite viscosity, the surface tension, the diffusion coefficient of
References
[1]
R. Keunings, “Advances in the computer modeling of the flow of polymeric liquids,” Computational Fluid Dynamics Journal, vol. 9, pp. 449–458, 2001.
[2]
L. Lefebvre and R. Keunings, “Numerical simulation of chemically-reacting polymer flows,” in Proceedings of the 1st European Computational Fluid Dynamics Conference, vol. 2, pp. 1133–1138, Brussels, Belgium, September 1992.
[3]
A. Arefmanesh and S. G. Advani, “Diffusion-induced growth of a gas bubble in a viscoelastic fluid,” Rheologica Acta, vol. 30, no. 3, pp. 274–283, 1991.
[4]
D. F. Baldwin, C. B. Park, and N. P. Suh, “Microcellular sheet extrusion system process design models for shaping and cell growth control,” Polymer Engineering & Science, vol. 38, no. 4, pp. 674–688, 1998.
[5]
R. Benzi, S. Succi, and M. Vergassola, “The lattice Boltzmann equation: theory and applications,” Physics Report, vol. 222, no. 3, pp. 145–197, 1992.
[6]
P. L. Bhatnagar, E. P. Gross, and M. Krook, “A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems,” Physical Review, vol. 94, no. 3, pp. 511–525, 1954.
[7]
J. H. Han and C. D. Han, “Bubble nucleation in polymeric liquids. I. Bubble nucleation in concentrated polymer solutions,” Journal of Polymer Science B, vol. 28, no. 5, pp. 711–741, 1990.
[8]
J. H. Han and C. D. Han, “Bubble nucleation in polymeric liquids. II. Theoretical considerations,” Journal of Polymer Science B, vol. 28, no. 5, pp. 743–761, 1990.
[9]
S. Hutzler, D. Weaire, and F. Bolton, “The effects of Plateau borders in the two-dimensional soap froth. III. Further results,” Philosophical Magazine B, vol. 71, no. 3, pp. 277–289, 1995.
[10]
C. Isenberg, The Science of Films and Soap Bubbles, Dover, New York, NY, USA, 1992.
[11]
D. Weaire and R. Phelan, “A counter-example to Kelvin's conjecture on minimal surfaces,” Philosophical Magazine Letters, vol. 69, no. 2, pp. 107–110, 1994.
[12]
J. U. Brackbill, D. B. Kothe, and C. Zemach, “A continuum method for modeling surface tension,” Journal of Computational Physics, vol. 100, no. 2, pp. 335–354, 1992.
[13]
N. S. Ramesh, D. H. Rasmussen, and G. A. Campbell, “Numerical and experimental studies of bubble growth during the microcellular foaming process,” Polymer Engineering & Science, vol. 31, no. 23, pp. 1657–1664, 1991.
[14]
L. E. Scriven, “On the dynamics of phase growth,” Chemical Engineering Science, vol. 10, no. 1-2, pp. 1–13, 1959.
[15]
D. L. Weaire and S. Hutzler, The Physics of Foams, Oxford University Press, New York, NY, USA, 1999.
[16]
R. Clift, J. R. Grace, and M. Weber, Bubbles, Drops and Particles, Academic Press, New York, NY, USA, 2002.
[17]
F. Bolton and D. Weaire, “The effects of Plateau borders in the two-dimensional soap froth I. Decoration lemma and diffusion theorem,” Philosophical Magazine B, vol. 63, no. 4, pp. 795–809, 1991.
[18]
J. M. Buick, Lattice Boltzmann methods in interfacial wave modelling [Ph.D. thesis], University of Edinburgh, Edinburgh, Scotland, 1997.
[19]
J. W. Bullard, E. J. Garboczi, W. C. Carter, and E. R. Fuller Jr., “Numerical methods for computing interfacial mean curvature,” Computational Materials Science, vol. 4, no. 2, pp. 103–116, 1995.
[20]
N. Cao, S. Chen, S. Jin, and D. Martinez, “Physical symmetry and lattice symmetry in the lattice Boltzmann method,” Physical Review E, vol. 55, no. 1, pp. R21–R28, 1997.
[21]
X. He and L.-S. Luo, “A priori derivation of the lattice Boltzmann equation,” Physical Review E, vol. 55, no. 6, pp. R6333–R6336, 1997.
[22]
X. He and L.-S. Luo, “Theory of the lattice Boltzmann method: from the Boltzmann equation to the lattice Boltzmann equation,” Physical Review E, vol. 56, no. 6, pp. 6811–6817, 1997.
[23]
Z. Guo, C. Zheng, and B. Shi, “Discrete lattice effects on the forcing term in the lattice Boltzmann method,” Physical Review E, vol. 65, no. 4, Article ID 046308, 6 pages, 2002.
[24]
J. M. Buick and C. A. Created, “Gravity in a lattice Boltzmann model,” Physical Review E, vol. 61, no. 5, pp. 5307–5320, 2000.
[25]
S. Chen, D. Martínez, and R. Mei, “On boundary conditions in lattice Boltzmann methods,” Physics of Fluids, vol. 8, no. 9, pp. 2527–2536, 1996.
[26]
M. R. Swift, E. Orlandini, W. R. Osborn, and J. M. Yeomans, “Lattice Boltzmann simulations of liquid-gas and binary fluid systems,” Physical Review E, vol. 54, no. 5, pp. 5041–5052, 1996.
[27]
A. K. Gunstensen, D. H. Rothman, S. Zaleski, and G. Zanetti, “Lattice Boltzmann model of immiscible fluids,” Physical Review A, vol. 43, no. 8, pp. 4320–4327, 1991.
[28]
D. Grunau, S. Chen, and K. Eggert, “A lattice Boltzmann model for multiphase fluid flows,” Physics of Fluids A, vol. 5, no. 10, pp. 2557–2562, 1993.
[29]
M. Do-Quang, E. Aurell, and M. Vergassola, “An inventory of lattice Boltzmann models of multiphase flows,” Tech. Rep., Parallel and Scientific Computing Institute, Department of Scientific Computing, Uppsala University, Uppsala, Sweden, 2000.
